Rossby-gravity waves

Rossby-gravity waves are equatorially trapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator (so long as the Brunt–Vaisala frequency does not remain constant). These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius,[1] they always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

Contents

The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H,[2] because the Coriolis parameter (f = 2Ω sin(θ) where Ω is the angular velocity of the earth, 7.2921 × 10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane” approximation must be made. This approximation states that f is approximately equal to βy, where y is the distance from the equator and β is the variation of the Coriolis parameter with latitude, ∂f∂y=β{\displaystyle {\frac {\partial f}{\partial y}}=\beta }.[3] With the inclusion of this approximation, the primitive equations become (neglecting friction):

the continuity equation (accounting for the effects of horizontal convergence and divergence and written with geopotential height):

These three equations can be separated and solved using solutions in the form of zonally propagating waves, which are analogous to exponential solutions with a dependence on x and t and the inclusion of structure functions that vary in the y-direction:

Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with three distinct solutions, these three solutions correspond to the equatorially trapped gravity wave, the equatorially trapped Rossby wave and the mixed Rossby-gravity wave (which has some of the characteristics of the former two) .[3] Equatorial gravity waves can be either westward- or eastward-propagating, and correspond to n=1 (same as for the equatorially trapped Rossby wave) on a dispersion relation diagram ("w-k" diagram), at n = 0 on a dispersion relation diagram, the mixed Rossby-gravity waves can be found where for large, positive zonal wave numbers (+k), the solution behaves like a gravity wave; but for large, negative zonal wave numbers (−k), the solution appears to be a Rossby wave (hence the term Rossby-gravity waves).[1] As mentioned earlier, the group velocity (or energy packet/dispersion) is always directed toward the east with a maximum for short waves (gravity waves).[1]

As previously stated, the mixed Rossby-gravity waves are equatorially trapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency. If this Brunt–Vaisala frequency does not change, then these waves become vertically propagating solutions,[1] on a typical "m,k" dispersion diagram, the group velocity (energy) would be directed at right angles to the n = 0 (mixed Rossby-gravity waves) and n = 1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency.[1] Typical group velocities for each component are the following: 1 cm/s for gravity waves and 2 mm/s for planetary (Rossby) waves.[1]

These vertically propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai,[4] they had the following characteristics: 4–5 days, horizontal wavenumbers of 4 (four waves circling the earth, corresponding to wavelengths of 10,000 km), vertical wavelengths of 4–8 km, and upward group velocity.[1] Similarly, westward-propagating mixed waves were also found in the Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200 km, vertical wavelengths of 1 km, and downward group velocity.[1] Also, the vertically propagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400 km, and vertical wavelengths of 5 km.[1]

1.
Kelvin waves
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A Kelvin wave is a wave in the ocean or atmosphere that balances the Earths Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive and this means that it retains its shape as it moves in the alongshore direction over time. This wave is named after the discoverer, Lord Kelvin, in a stratified ocean of mean depth H, free waves propagate along coastal boundaries in the form of internal Kelvin waves on a scale of about 30 km. These waves are called coastal Kelvin waves, and have speeds of approximately 2 m/s in the ocean. The solution to these equations yields the following phase speed, c2 = gH and it is important to note that for an observer traveling with the wave, the coastal boundary is always to the right in the northern hemisphere and to the left in the southern hemisphere. The primitive equations are identical to those used to develop the coastal Kelvin wave phase speed solution and this equatorial Beta plane assumption requires a geostrophic balance between the eastward velocity and the north-south pressure gradient. The phase speed is identical to that of coastal Kelvin waves, for the first baroclinic mode in the ocean, a typical phase speed would be about 2. There have been studies that connect equatorial Kelvin waves to coastal Kelvin waves and this process indicates that some energy may be lost from the equatorial region and transported to the poleward region. Equatorial Kelvin waves are associated with anomalies in surface wind stress. For example, positive anomalies in wind stress in the central Pacific excite positive anomalies in 20°C isotherm depth which propagate to the east as equatorial Kelvin waves, rossby wave Rossby-gravity waves Kelvin-Helmholtz instability Edge wave Overview of Kelvin waves from the American Meteorological Society. US Navy page on Kelvin waves, slideshow at utexus. edu about Kelvin waves. Kelvin Wave Renews El Niño - NASA, Earth Observatory, image of the day,2010 March 21

2.
Angular velocity
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This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega, the direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule. The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram, if a line is drawn from the origin to the particle, then the velocity of the particle has a component along the radius, if there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, a radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, the angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis, if the parity is inverted, but the orientation of a rotation is not, then the sign of the angular velocity changes. There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in case is generally thought of as a vector, or more precisely. It now has not only a magnitude, but a direction as well, the magnitude is the angular speed, and the direction describes the axis of rotation that Eulers rotation theorem guarantees must exist. The right-hand rule indicates the direction of the angular velocity pseudovector. Let u be a vector along the instantaneous rotation axis. This is the definition of a vector space, the only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W is skew-symmetric, therefore, R = e W t is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore, it can be expanded as R = I + W ⋅ d t +122 +, in such a frame, each vector is a particular case of the previous case, in which the module of the vector is constant. Though it just a case of a moving particle, this is a very important one for its relationship with the rigid body study. There are two ways to describe the angular velocity of a rotating frame, the angular velocity vector. Both entities are related and they can be calculated from each other, in a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame

3.
Rossby wave
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Rossby waves, also known as planetary waves, are a natural phenomenon in the atmosphere and oceans of planets that largely owe their properties to rotation of the planet. Rossby waves are a subset of inertial waves, atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. These waves are associated with systems and the jet stream. Oceanic Rossby waves move along the thermocline, the boundary between the upper layer and the cold deeper part of the ocean. Atmospheric Rossby waves result from the conservation of vorticity and are influenced by the Coriolis force. A fluid, on the Earth, that moves toward the pole will deviate toward the east, the deviations are caused by the Coriolis force and conservation of potential vorticity which leads to changes of relative vorticity. This is analogous to conservation of momentum in mechanics. In planetary atmospheres, including Earth, Rossby waves are due to the variation in the Coriolis effect with latitude, carl-Gustaf Arvid Rossby first identified such waves in the Earths atmosphere in 1939 and went on to explain their motion. One can identify a terrestrial Rossby wave as its velocity, marked by its wave crest. However, the set of Rossby waves may appear to move in either direction with what is known as its group velocity. In general, shorter waves have a group velocity and long waves a westward group velocity. The terms barotropic and baroclinic are used to distinguish the structure of Rossby waves. Barotropic Rossby waves do not vary in the vertical, and have the fastest propagation speeds, the baroclinic wave modes, on the other hand, do vary in the vertical. They are also slower, with speeds of only a few centimeters per second or less, most investigations of Rossby waves has been done on those in Earths atmosphere. Rossby waves in the Earths atmosphere are easy to observe as large-scale meanders of the jet stream, the action of Rossby waves partially explains why eastern continental edges, such as the Northeast United States and Eastern Canada, are colder than Western Europe at the same latitudes. Deep convection to the troposphere is enhanced over very warm sea surfaces in the tropics and this tropical forcing generates atmospheric Rossby waves that have a poleward and eastward migration. Poleward-propagating Rossby waves explain many of the observed statistical connections between low- and high-latitude climates, one such phenomenon is sudden stratospheric warming. Poleward-propagating Rossby waves are an important and unambiguous part of the variability in the Northern Hemisphere, similar mechanisms apply in the Southern Hemisphere and partly explain the strong variability in the Amundsen Sea region of Antarctica

4.
Physical oceanography
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Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters. Physical oceanography is one of several sub-domains into which oceanography is divided, others include biological, chemical and geological oceanography. Physical oceanography may be subdivided into descriptive and dynamical physical oceanography, descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precise as possible. Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and these are part of the large field of Geophysical Fluid Dynamics that is shared together with meteorology. The fundamental role of the oceans in shaping Earth is acknowledged by ecologists, geologists, meteorologists, climatologists, an Earth without oceans would truly be unrecognizable. Roughly 97% of the water is in its oceans. The tremendous heat capacity of the oceans moderates the planets climate, the oceans influence extends even to the composition of volcanic rocks through seafloor metamorphism, as well as to that of volcanic gases and magmas created at subduction zones. Though this apparent discrepancy is great, for land and sea, the respective extremes such as mountains and trenches are rare. Because the vast majority of the oceans volume is deep water. The same percentage falls in a salinity range between 34–35 ppt, there is still quite a bit of variation, however. Surface temperatures can range from below freezing near the poles to 35 °C in restricted tropical seas, in terms of temperature, the oceans layers are highly latitude-dependent, the thermocline is pronounced in the tropics, but nonexistent in polar waters. The halocline usually lies near the surface, where evaporation raises salinity in the tropics and these variations of salinity and temperature with depth change the density of the seawater, creating the pycnocline. Energy for the ocean circulation comes from solar radiation and gravitational energy from the sun, perhaps three quarters of this heat is carried in the atmosphere, the rest is carried in the ocean. The atmosphere is heated from below, which leads to convection, by contrast the ocean is heated from above, which tends to suppress convection. Instead ocean deep water is formed in regions where cold salty waters sink in fairly restricted areas. This is the beginning of the thermohaline circulation, oceanic currents are largely driven by the surface wind stress, hence the large-scale atmospheric circulation is important to understanding the ocean circulation. The Hadley circulation leads to Easterly winds in the tropics and Westerlies in mid-latitudes and this leads to slow equatorward flow throughout most of a subtropical ocean basin. The return flow occurs in an intense, narrow, poleward western boundary current, like the atmosphere, the ocean is far wider than it is deep, and hence horizontal motion is in general much faster than vertical motion

5.
Wind wave
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In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the free surface of bodies of water. They result from the wind blowing over an area of fluid surface, Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft high, when directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells, more generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago, wind waves in the ocean are called ocean surface waves. Wind waves have an amount of randomness, subsequent waves differ in height, duration. The key statistics of wind waves in evolving sea states can be predicted with wind wave models, although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves. The great majority of large breakers seen at a result from distant winds. Water depth All of these work together to determine the size of wind waves. Further exposure to that wind could only cause a dissipation of energy due to the breaking of wave tops. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer would estimate from visual observation of a sea state, given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. Wave formation on a flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water and it is assumed that, The water is originally at rest. There is a distribution of normal pressure to the water surface from the turbulent wind. Correlations between air and water motions are neglected, the second mechanism involves wind shear forces on the water surface

6.
Airy wave theory
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In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the layer has a uniform mean depth. This theory was first published, in form, by George Biddell Airy in the 19th century. Further, several second-order nonlinear properties of gravity waves, and their propagation. Airy wave theory is also a good approximation for tsunami waves in the ocean and this linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the height to water depth. Airy wave theory uses a potential approach to describe the motion of gravity waves on a fluid surface. This is due to the fact that for the part of the fluid motion. Airy wave theory is used in ocean engineering and coastal engineering. Diffraction is one of the effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoaling and refraction can be predicted, earlier attempts to describe surface gravity waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841, soon after, in 1847, the linear theory of Airy was extended by Stokes for non-linear wave motion – known as Stokes wave theory – correct up to third order in the wave steepness. Even before Airys linear theory, Gerstner derived a nonlinear wave theory in 1802. Airy wave theory is a theory for the propagation of waves on the surface of a potential flow. The waves propagate along the surface with the phase speed cp. The angular wavenumber k and frequency ω are not independent parameters, surface gravity waves on a fluid are dispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed. Note that in engineering the wave height H – the difference in elevation between crest and trough – is often used, H =2 a and a =12 H, underneath the surface, there is a fluid motion associated with the free surface motion. While the surface shows a propagating wave, the fluid particles are in an orbital motion

7.
Ballantine scale
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The Ballantine scale is a biologically defined scale for measuring the degree of exposure level of wave action on a rocky shore. The species present in the littoral zone therefore indicate the degree of the shores exposure, an abbreviated summary of the scale is given below. The scale runs from 1) an extremely exposed shore, to 8) an extremely sheltered shore, the littoral zone generally is the zone between low and high tides. The supra-littoral is above the barnacle line, the eulittoral zone is dominated by barnacles and limpets with a coralline belt in the very low littoral along with other Rhodophyta and Alaria in the upper sublittoral. Exposed shores show a Verrucaria belt mainly above the tide, with Porphyra. The mid shore is dominated by barnacles, limpets and some Fucus, Himanthalia and some Rhodophyta such as Mastocarpus and Corallina are found in the low littorral with Himanthalia, Alaria and Laminaria digitata dominant in the upper sublittoral. Semi-exposed shores show a Verrucaria belt just above high tide with clear Pelvetia in the upper-littoral, limpets, barnacles and short Fucus vesiculosus midshore. Laminaria and Saccorhiza polyschides and small algae common in the sublittoral, sheltered shores show a narrow Verrucaria zone at high water and a full sequence of fucoids, Pelvetia, Fucus spiralis, Fucus vesiculosus, Fucus serratus, Ascophyllum nodosum. Laminaria digitata is dominant the upper sublittoral, very sheltered shores show a very narrow zone of Verrucaria, the dominance of the littoral by a full sequence of the fucoids and Ascophyllum covering the rocks. Laminaria saccharina, Halidrys, Chondrus and or Furcellaria, a Biologically-defined Exposure Scale for the Comparative Description of Rocky Shores

8.
Modulational instability
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The phenomenon was first discovered − and modelled − for periodic surface gravity waves on deep water by T. Brooke Benjamin and Jim E. Feir, in 1967. Therefore, it is known as the Benjamin−Feir instability. It is a mechanism for the generation of rogue waves. Modulation instability only happens under certain circumstances, the most important condition is anomalous group velocity dispersion, whereby pulses with shorter wavelengths travel with higher group velocity than pulses with longer wavelength. The instability is strongly dependent on the frequency of the perturbation, at certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will grow exponentially. The overall gain spectrum can be derived analytically, as is shown below, random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum. The tendency of a signal to grow makes modulation instability a form of amplification. By tuning an input signal to a peak of the gain spectrum, the imaginary unit i satisfies i 2 = −1. The model includes group velocity dispersion described by the parameter β2, a periodic waveform of constant power P is assumed. The beginning of instability can be investigated by perturbing this solution as A = e i γ P z, the complex conjugate of ε is denoted as ε ∗. Instability can now be discovered by searching for solutions of the equation which grow exponentially. The nonlinear Schrödinger equation is constructed by removing the carrier wave of the light being modelled, therefore, ω m and k m dont represent absolute frequencies and wavenumbers, but the difference between these and those of the initial beam of light. It can be shown that the function is valid, provided c 2 = c 1 ∗. Therefore, instability will occur when β22 ω m 2 +2 γ P β2 <0 and this condition describes the requirement for anomalous dispersion. The gain spectrum can be described by defining a gain parameter as g ≡2 | ℑ |, the growth rate is maximum for ω2 = − γ P / β2. Modulation instability of optical fields has been observed in systems, namely. Modulation instability occurs owing to inherent optical nonlinearity of the due to photoreaction-induced changes in the refractive index. Ostrovsky, L. A. Modulation instability, The beginning

9.
Boussinesq approximation (water waves)
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In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation, the 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the structure of the horizontal and vertical flow velocity. This results in partial differential equations, called Boussinesq-type equations. In coastal engineering, Boussinesq-type equations are used in computer models for the simulation of water waves in shallow seas. While the Boussinesq approximation is applicable to fairly long waves – that is and this is useful because the waves propagate in the horizontal plane and have a different behaviour in the vertical direction. Often, as in Boussinesqs case, the interest is primarily in the wave propagation and this elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave. Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations, thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the partial differential equations are in terms of functions of the horizontal coordinates. As an example, consider potential flow over a bed in the plane, with x the horizontal. The bed is located at z = −h, where h is the water depth. Invoking Laplaces equation for φ, as valid for incompressible flow, gives, φ = + = and this series may subsequently be truncated to a finite number of terms. Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions, further, in the resulting equations only the linear and quadratic terms with respect to η and ub are retained. The cubic and higher terms are assumed to be negligible. This set of equations has been derived for a horizontal bed. When the right-hand sides of the equations are set to zero. From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number, for the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation

10.
Breaking wave
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At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour. The most generally familiar sort of breaking wave is the breaking of surface waves on a coastline. Wave breaking generally occurs where the amplitude reaches the point that the crest of the wave actually overturns—though the types of breaking water waves are discussed in more detail below. Certain other effects in fluid dynamics have also been termed breaking waves, wave breaking also occurs in plasmas, when the particle velocities exceed the waves phase speed. Breaking of water surface waves may occur anywhere that the amplitude is sufficient, however, it is particularly common on beaches because wave heights are amplified in the region of shallower water. See also waves and shallow water, there are four basic types of breaking water waves. They are spilling, plunging, collapsing, and surging, when the ocean floor has a gradual slope, the wave will steepen until the crest becomes unstable, resulting in turbulent whitewater spilling down the face of the wave. This continues as the approaches the shore, and the waves energy is slowly dissipated in the whitewater. Because of this, spilling waves break for a time than other waves. Onshore wind conditions make spillers more likely, a plunging wave occurs when the ocean floor is steep or has sudden depth changes, such as from a reef or sandbar. A plunging wave breaks with more energy than a significantly larger spilling wave, the wave can trap and compress the air under the lip, which creates the crashing sound associated with waves. With large waves, this crash can be felt by beachgoers on land, offshore wind conditions can make plungers more likely. This is the tube that is so highly sought after by surfers, the surfer tries to stay near or under the crashing lip, often trying to stay as deep in the tube as possible while still being able to shoot forward and exit the barrel before it closes. A plunging wave that is parallel to the beach can break along its length at once, rendering it unrideable. Surfers refer to waves as closed out. Collapsing waves are a cross between plunging and surging, in which the crest never fully breaks, yet the bottom face of the wave gets steeper and collapses, surging breakers originate from long period, low steepness waves and/or steep beach profiles. The outcome is the movement of the base of the wave up the swash slope. The front face and crest of the wave remain relatively smooth with little foam or bubbles, resulting in a very narrow surf zone, or no breaking waves at all

11.
Clapotis
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The resulting clapotic wave does not travel horizontally, but has a fixed pattern of nodes and antinodes. These waves promote erosion at the toe of the wall, the term was coined in 1877 by French mathematician and physicist Joseph Valentin Boussinesq who called these waves ‘le clapotis’ meaning ‘’the lapping. The standing waves alternately rise and fall in a mirror image pattern, as energy is converted to potential energy. This may also occur at sea between two different wave trains of near equal wavelength moving in opposite directions, but with unequal amplitudes, in partial clapotis the wave envelope contains some vertical motion at the nodes. When a wave strikes a wall at an oblique angle. In this situation, the individual crests formed at the intersection of the incident and this wave motion, when combined with the resultant vortices, can erode material from the seabed and transport it along the wall, undermining the structure until it fails. Clapotic waves on the sea surface also radiate infrasonic microbaroms into the atmosphere, clapotis has been called the bane and the pleasure of Sea kayaking. Rogue wave Boussinesq, J. Théorie des ondes liquides périodiques, mémoires présentés par divers savants à lAcadémie des Sciences. Boussinesq, J. Essai sur la théorie des eaux courantes, mémoires présentés par divers savants à lAcadémie des Sciences. Clapotis and Wave Reflection, With an Application to Vertical Breakwater Design, clapotis Wave Action – via YouTube

12.
Cnoidal wave
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In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn and they are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth. The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, in the limit of infinite wavelength, the cnoidal wave becomes a solitary wave. The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. The KdV equation is a wave equation, including both frequency dispersion and amplitude dispersion effects. Shallow water equations — are also nonlinear and do have amplitude dispersion, Boussinesq equations — have the same range of validity as the KdV equation, but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are more difficult to solve than the KdV equation. Airy wave theory — has full frequency dispersion, so valid for arbitrary depth and wavelength, however, for long waves the Boussinesq approach—as also applied in the KdV equation—is often preferred. This is because in shallow water the Stokes perturbation series needs many terms before convergence towards the solution, due to the peaked crests, while the KdV or Boussinesq models give good approximations for these long nonlinear waves. The KdV equation can be derived from the Boussinesq equations, further improvements in short-wave performance can be obtained by starting to derive a one-way wave equation from a modern improved Boussinesq model, valid for even shorter wavelengths. The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, solitary wave solutions for nonlinear and dispersive long waves had been found earlier by Boussinesq in 1872, and Rayleigh in 1876. The search for these solutions was triggered by the observations of this solitary wave by Russell, cnoidal wave solutions of the KdV equation are stable with respect to small perturbations. Further cn is one of the Jacobi elliptic functions and K is the elliptic integral of the first kind. The latter, m, determines the shape of the cnoidal wave, for m equal to zero the cnoidal wave becomes a cosine function, while for values close to one the cnoidal wave gets peaked crests and flat troughs. For values of m less than 0.95, the function can be approximated with trigonometric functions. An important dimensionless parameter for nonlinear long waves is the Ursell parameter, for small values of U, say U <5, a linear theory can be used, and at higher values nonlinear theories have to be used, like cnoidal wave theory. The demarcation zone between—third or fifth order—Stokes and cnoidal wave theories is in the range 10–25 of the Ursell parameter. Based on the analysis of the nonlinear problem of surface gravity waves within potential flow theory

13.
Cross sea
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In surface navigation, a cross sea is a sea state with two wave systems traveling at oblique angles. This may occur when waves from one weather system continue despite a shift in wind. Waves generated by the new wind run at an angle to the old, creating a shifting, two weather systems that are far from each other may create a cross sea when the waves from the systems meet, usually at a place far from either weather system. Until the older waves have dissipated, they create a sea hazard among the most perilous and this sea state is fairly common and a larger percentage of ship accidents were found to have occurred in this state. A cross swell is generated when the systems are longer period swell

14.
Dispersion (water waves)
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In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in context, are waves propagating on the water surface, with gravity. As a result, water with a surface is generally considered to be a dispersive medium. For a certain depth, surface gravity waves – i. e. waves occurring at the air–water interface. On the other hand, for a wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves propagate faster for shorter wavelengths, besides frequency dispersion, water waves also exhibit amplitude dispersion. This is an effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves. This section is about frequency dispersion for waves on a fluid layer forced by gravity, for surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave. The simplest propagating wave of unchanging form is a sine wave. Characteristic phases of a wave are, the upward zero-crossing at θ =0, the wave crest at θ = ½ π, the downward zero-crossing at θ = π. A certain phase repeats itself after an integer m multiple of 2π, the dispersion relation has two solutions, ω = +Ω and ω = −Ω, corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on other parameters in addition to the wavenumber k. For gravity waves, according to theory, these are the acceleration by gravity g. The dispersion relation for these waves is, an equation with tanh denoting the hyperbolic tangent function. An initial wave phase θ = θ0 propagates as a function of space and its subsequent position is given by, x = λ T t + λ2 π θ0 = ω k t + θ0 k. This shows that the moves with the velocity, c p = λ T = ω k = Ω k. A sinusoidal wave, of small amplitude and with a constant wavelength, propagates with the phase velocity. While the phase velocity is a vector and has an associated direction, according to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth

15.
Equatorial wave
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Equatorial waves are ocean waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earths rotation and its shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Equatorial waves may be separated into a series of subclasses depending on their fundamental dynamics, at shortest periods are the equatorial gravity waves while the longest periods are associated with the equatorial Rossby waves. In addition to these two subclasses, there are two special subclasses of equatorial waves known as the mixed Rossby-gravity wave and the equatorial Kelvin wave. The latter two share the characteristics that they can have any period and also that they may carry only in an eastward direction. The remainder of this article discusses the relationship between the period of waves, their wavelength in the zonal direction and their speeds for a simplified ocean. Rossby-gravity waves, first observed in the stratosphere by M. Yanai, but, oddly, their crests and troughs may propagate westward if their periods are long enough. The eastward speed of propagation of waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter vanishes at 0 degrees latitude, the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and β is the variation of the coriolis parameter with latitude, ∂ f ∂ y = β. Once the frequency relation is formulated in terms of ω, the angular frequency and these three solutions correspond to the equatorial gravity waves, the equatorially trapped Rossby waves and the mixed Rossby-gravity wave. Equatorial gravity waves can be either westward- or eastward-propagating, the governing equations for these equatorial waves are similar to those presented above, except that there is no meridional velocity component. The continuity equation, ∂ ϕ ∂ t + c 2 ∂ u ∂ x =0 the u-momentum equation, ∂ u ∂ t = − ∂ ϕ ∂ x the v-momentum equation, u β y = − ∂ ϕ ∂ y. The solution to these equations yields the following phase speed, c2 = gH, also, these Kelvin waves only propagate towards the east. Kelvin waves have been connected to El Niño in recent years in terms of precursors to this atmospheric and oceanic phenomenon, the weak low pressure in the Indian Ocean typically propagates eastward into the North Pacific Ocean and can produce easterly winds. This wave can be observed at the surface by a rise in sea surface height of about 8 cm. If the Kelvin wave hits the South American coast, its water gets transferred upward

16.
Fetch (geography)
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The fetch, also called the fetch length, is the length of water over which a given wind has blown. It also plays a part in longshore drift as well. Fetch length, along with the speed, determines the size of waves produced. The wind direction is considered constant, the longer the fetch and the faster the wind speed, the more wind energy is imparted to the water surface and the larger the resulting sea state will be. Sea state Ocean surface wave Storm surge

17.
Gravity wave
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In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which rise to wind waves. A gravity wave results when fluid is displaced from a position of equilibrium, the restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a wave orbit. Gravity waves on an interface of the ocean are called surface gravity waves or surface waves. Wind-generated waves on the surface are examples of gravity waves, as are tsunamis. Wind-generated gravity waves on the surface of the Earths ponds, lakes, seas. Shorter waves are affected by surface tension and are called gravity–capillary waves. Alternatively, so-called infragravity waves, which are due to nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves. In the Earths atmosphere, gravity waves are a mechanism that produce the transfer of momentum from the troposphere to the stratosphere and mesosphere, Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first, waves propagate through the atmosphere without appreciable change in mean velocity, but as the waves reach more rarefied air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere, thus, this process plays a key role in the dynamics of the middle atmosphere. The effect of gravity waves in clouds can look like altostratus undulatus clouds, and are confused with them. The phase velocity c of a gravity wave with wavenumber k is given by the formula c = g k. When surface tension is important, this is modified to c = g k + σ k ρ, where σ is the surface tension coefficient and ρ is the density. Since c = ω / k is the speed in terms of the angular frequency ω and the wavenumber. The group velocity of a wave is given by c g = d ω d k, the group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive, Gravity waves traveling in shallow water, are nondispersive, the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is h, c p = c g = g h, wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the oceans surface, and capillary-gravity waves play an essential role in this effect

18.
Infragravity wave
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Infragravity waves are ocean surface gravity waves generated by ocean waves of shorter periods. The amplitude of infragravity waves is most relevant in shallow water, in particular along coastlines hit by high amplitude and long period wind waves, wind waves and ocean swells are shorter, with typical dominant periods of 1 to 25 s. This distinguishes infragravity waves from normal oceanic gravity waves, which are created by wind acting on the surface of the sea, whatever the details of their generation mechanism, discussed below, infragravity waves are these subharmonics of the impinging gravity waves. Technically infragravity waves are simply a subcategory of gravity waves and refer to all gravity waves with greater than 30 s. This could include such as tides and oceanic Rossby waves. The term infragravity wave appears to have coined by Walter Munk in 1950. Two main processes can explain the transfer of energy from the wind waves to the long infragravity waves. The most common process is the interaction of trains of wind waves which was first observed by Munk and Tucker and explained by Longuet-Higgins. Because wind waves are not monochromatic they form groups, the Stokes drift induced by these groupy waves transports more water where the waves are highest. The waves also push the water around in a way that can be interpreted as a force, combining mass and momentum conservation, Longuet-Higgins and Stewart give, with three different methods, the now well-known result. Namely, the sea level oscillates with a wavelength that is equal to the length of the group, with a low level where the wind waves are highest. This oscillation of the sea surface is proportional to the square of the wave amplitude. Another process was proposed later by Graham Symonds and his collaborators and it appears that this is probably a good explanation for infragravity wave generation on a reef. In the case of coral reefs, the infragravity periods are established by resonances with the reef itself, infragravity waves generated along the Pacific coast of North America have been observed to propagate transoceanically to Antarctica and there to impinge on the Ross Ice Shelf. Their frequencies more closely couple with the ice shelf natural frequencies, further, they are not damped by sea ice as normal ocean swell is. As a result, they flex floating ice shelves such as the Ross Ice Shelf, media related to Gravity waves at Wikimedia Commons

19.
Internal wave
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Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified, the density must decrease continuously or discontinuously with depth/height due to changes, for example, If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid. Internal waves, also called gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude. If propagating horizontally along an interface where the density decreases with height. If the interfacial waves are large amplitude they are called solitary waves or internal solitons. If moving vertically through the atmosphere where substantial changes in air density influences their dynamics, If generated by flow over topography, they are called Lee waves or mountain waves. If the mountain waves break aloft, they can result in strong winds at the ground known as Chinook winds or Foehn winds. If generated in the ocean by tidal flow over submarine ridges or the continental shelf, If they evolve slowly compared to the Earths rotational frequency so that their dynamics are influenced by the Coriolis effect, they are called inertia gravity waves or, simply, inertial waves. Internal waves are usually distinguished from Rossby waves, which are influenced by the change of Coriolis frequency with latitude. An internal wave can readily be observed in the kitchen by slowly tilting back, clouds that reveal internal waves launched by flow over hills are called lenticular clouds because of their lens-like appearance. Less dramatically, a train of waves can be visualized by rippled cloud patterns described as herringbone sky or mackerel sky. The outflow of air from a thunderstorm can launch large amplitude internal solitary waves at an atmospheric inversion. In northern Australia, these result in Morning Glory clouds, used by some daredevils to glide along like a surfer riding an ocean wave, satellites over Australia and elsewhere reveal these waves can span many hundreds of kilometers. According to Archimedes principle, the weight of an object is reduced by the weight of fluid it displaces. This holds for a parcel of density ρ surrounded by an ambient fluid of density ρ0. Its weight per volume is g, in which g is the acceleration of gravity. Dividing by a density, ρ00, gives the definition of the reduced gravity. Because water is more dense than air, the displacement of water by air from a surface gravity wave feels nearly the full force of gravity

20.
Kinematic wave
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These waves are also applied to model the motion of highway traffic flows. In these flows, mass and momentum equations can be combined to yield a kinematic wave equation, depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. In general, the wave can be advecting and diffusing, however, in simple situation, the kinematic wave is mainly advecting. For F = h 2 /2, this reduces to the Burgers equation

21.
Longshore drift
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Longshore drift is a geological process that consists of the transportation of sediments along a coast parallel to the shoreline, which is dependent on oblique incoming wind direction. Oblique incoming wind squeezes water along the coast, and so generates a current which moves parallel to the coast. Longshore drift is simply the sediment moved by the longshore current and this current and sediment movement occurs within the surf zone. Beach sand is moved on such oblique wind days, due to the swash and backwash of water on the beach. Breaking surf sends water up the beach at an oblique angle, thus beach sand can move downbeach in a zig zag fashion many tens of meters per day. This process is called beach drift but some regard it as simply part of longshore drift because of the overall movement of sand parallel to the coast. Longshore drift affects numerous sediment sizes as it works in different ways depending on the sediment. Sand is largely affected by the force of breaking waves. There are numerous calculations that take into consideration the factors that produce longshore drift, some of these are, Geological changes, e. g. erosion, backshore changes and emergence of headlands. Change in hydrodynamic forces, e. g. change in wave diffraction in headland, change to hydrodynamic influences, e. g. the influence of new tidal inlets and deltas on drift. Alterations of the sediment budget, e. g. switch of shorelines from drift to swash alignment, the intervention of humans, e. g. cliff protection, groynes, detached breakwaters. The sediment budget takes into consideration sediment sources and sinks within a system, a good example of the sediment budget and longshore drift working together in the coastal system is inlet ebb-tidal shoals, which store sand that has been transported by long shore transport. As well as storing sand these systems may also transfer or by pass sand into other systems, therefore inlet ebb-tidal systems provide a good sources. Long shore occurs in a 90 to 80 degree backwash so it would be presented as an angle with the wave line. This section consists of features of long shore drift that occur on a coast where long shore drift occurs uninterrupted by man-made structures, spits are formed when longshore drift travels past a point where the dominant drift direction and shoreline do not veer in the same direction. As well as dominant drift direction, spits are affected by the strength of wave driven current, wave angle, spits are landforms that have two important features, with the first feature being the region at the up-drift end or proximal end. The proximal end is attached to land and may form a slight “barrier” between the sea and an estuary or lagoon. As an example, the New Brighton spit in Canterbury, New Zealand, was created by longshore drift of sediment from the Waimakariri River to the north and this spit system is currently in equilibrium but undergoes phases of deposition and erosion

22.
Luke's variational principle
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In fluid dynamics, Lukes variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J. C. Luke, who published it in 1967, Lukes Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. This is often used when modelling the spectral density evolution of the free-surface in a sea state, both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity. Lukes Lagrangian formulation is for surface gravity waves on an—incompressible. The Lagrangian L, as given by Luke, is, L = − ∫ t 0 t 1 d t, from Bernoullis principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain V. This is in agreement with the principles for inviscid flow without a free surface. This may also include moving wavemaker walls and ship motion. For the case of an unbounded domain with the free fluid surface at z=η. The bed-level term proportional to h2 in the energy has been neglected, since it is a constant. Below, Lukes variational principle is used to arrive at the equations for non-linear surface gravity waves on a potential flow. The variation δ L =0 in the Lagrangian with respect to variations in the velocity potential Φ, consider a small variation δΦ in the velocity potential Φ. Then the resulting variation in the Lagrangian is, δ Φ L = L − L = − ∫ t 0 t 1 ∬ d x d t. The first integral on the right-hand side integrates out to the boundaries, in x and t, of the integration domain and is zero since the variations δΦ are taken to be zero at these boundaries. Similarly, variations δΦ only non-zero at the bottom z = -h result in the kinematic bed condition, ∇ Φ ⋅ ∇ h + ∂ Φ ∂ z =0 at z = − h. Considering the variation of the Lagrangian with respect to small changes δη gives, the Hamiltonian structure of surface gravity waves on a potential flow was discovered by Vladimir E. The Hamiltonian H is the sum of the kinetic and potential energy of the fluid and this is expressed by the Dirichlet-to-Neumann operator D, acting linearly on φ. The Hamiltonian density can also be written as, H =12 ρ φ +12 ρ g η2, as a result, the Hamiltonian is a quadratic functional of the surface potential φ. Also the potential energy part of the Hamiltonian is quadratic, the source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape η

23.
Mild-slope equation
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It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is used in coastal engineering to compute the wave-field changes near harbours. As a result, it describes the variations in wave amplitude, from the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. Most often, the equation is solved by computer using methods from numerical analysis. Also parabolic approximations to the equation are often used, in order to reduce the computational cost. In case of a constant depth, the equation reduces to the Helmholtz equation for wave diffraction. For a given angular frequency ω, the k has to be solved from the dispersion equation. The last equation shows that energy is conserved in the mild-slope equation. The effective group speed | v g | is different from the speed c g. The first equation states that the effective wavenumber κ is irrotational, a consequence of the fact it is the derivative of the wave phase θ. The second equation is the eikonal equation, otherwise, κ2 can even become negative. When the diffraction effects are neglected, the effective wavenumber κ is equal to k. The mild-slope equation can be derived by the use of several methods, here, we will use a variational approach. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational and these assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers. Because the flow is irrotational, the motion can be described using potential flow theory. The time-dependent mild-slope equation can be used to model waves in a band of frequencies around ω0. Consider monochromatic waves with complex amplitude η and angular frequency ω, ζ = ℜ, with ω and ω0 chosen equal to each other, ω = ω0. Using this in the time-dependent form of the equation, recovers the classical mild-slope equation for time-harmonic wave motion

24.
Radiation stress
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The radiation stresses behave as a second-order tensor. The radiation stress tensor describes the additional forcing due to the presence of the waves, as a result, varying radiation stresses induce changes in the mean surface elevation and the mean flow. For the mean energy density in the part of the fluid motion. Radiation stress derives its name from the effect of radiation pressure for electromagnetic radiation. As a result, a long wave propagates together with the group. While, according to the relation, a long wave of this length should propagate at its own – higher – phase velocity. For uni-directional wave propagation – say in the x-coordinate direction – the component of the stress tensor of dynamical importance is Sxx. Further ρ is the density and g is the acceleration by gravity. The last term on the side, ½ρg2, is the integral of the hydrostatic pressure over the still-water depth. Further E is the mean depth-integrated wave energy density per unit of horizontal area, note this equation is for periodic waves, in random waves the root-mean-square wave height Hrms should be used with Hrms = Hm0 / √2, where Hm0 is the significant wave height. For wave propagation in two dimensions the radiation stress S is a second-order tensor with components, S =. The phase and group speeds, cp and cg respectively, are the lengths of the phase and group velocity vectors, cp = |cp|, the radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Propagating waves induce a – relatively small – mean mass transport in the propagation direction. To lowest order, the wave momentum Mw is, per unit of area, M w = k k E c p. The difference k⋅v is the Doppler shift, the mean horizontal momentum M is also the mean of the depth-integrated horizontal mass flux, and consists of two contributions, one by the mean current and the other is due to the waves. Now the mass transport velocity u is defined as, u ¯ = M ρ = v ¯ + M w ρ, observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth is made. The equation of mass conservation is, in vector notation, ∂ ∂ t + ∇ ⋅ =0. Further I is the identity tensor, with components given by the Kronecker delta δij, note that the right hand side of the momentum equation provides the non-conservative contributions of the bed slope ∇h, as well the forcing by the wind and the bed friction

25.
Rogue wave
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Rogue waves are large, unexpected and suddenly appearing surface waves that can be extremely dangerous, even to large ships such as ocean liners. Rogue waves present considerable danger for several reasons, they are rare, unpredictable, may appear suddenly or without warning, a 12-metre wave in the usual linear model would have a breaking force of 6 metric tons per square metre. Although modern ships are designed to tolerate a breaking wave of 15 t/m2, therefore, rogue waves are not necessarily the biggest waves found on the water, they are, rather, unusually large waves for a given sea state. Rogue waves seem not to have a single cause, but occur where physical factors such as high winds. Rogue waves can occur in other than water. They appear to be ubiquitous in nature and have also reported in liquid helium, in nonlinear optics. Recent research has focused on optical rogue waves which facilitate the study of the phenomenon in the laboratory, once considered mythical and lacking hard evidence for their existence, rogue waves are now proven to exist and known to be a natural ocean phenomenon. Eyewitness accounts from mariners and damage inflicted on ships have long suggested they occurred, the first scientific evidence of the existence of rogue waves came with the recording of a rogue wave by the Gorm platform in the central North Sea in 1984. A stand-out wave was detected with a height of 11 metres in a relatively low sea state.6 metres. During that event, minor damage was inflicted on the platform, far above sea level, confirming that the reading was valid.5 metres. In 2004 scientists using three weeks of radar images from European Space Agency satellites found ten rogue waves, each 25 metres or higher. A rogue wave is a natural phenomenon that is not caused by land movement, only lasts briefly, occurs in a limited location. Rogue waves are distinct from tsunamis. Tsunamis are caused by displacement of water, often resulting from sudden movement of the ocean floor. They are also distinct from megatsunamis, which are single massive waves caused by sudden impact, Rogue waves have now been proven to be the cause of the sudden loss of some ocean-going vessels. Well documented instances include the freighter MS München, lost in 1978 and the MV Derbyshire lost in 1980, the largest British ship ever lost at sea. A rogue wave has been implicated in the loss of other vessels including the Ocean Ranger which was a mobile offshore drilling unit that sank in Canadian waters on 15 February 1982. In 2007 the US National Oceanic and Atmospheric Administration compiled a catalogue of more than 50 historical incidents probably associated with rogue waves, in that era it was widely held that no wave could exceed 30 feet

26.
Sea state
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In oceanography, a sea state is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterized by statistics, including the height, period. The sea state varies with time, as the conditions or swell conditions change. The sea state can either be assessed by an observer, like a trained mariner, or through instruments like weather buoys. In case of measurements, the statistics are determined for a time interval in which the sea state can be considered to be constant. This duration has to be longer than the individual wave period. Typically, records of one hundred to one thousand wave-periods are used to determine the wave statistics, the WMO sea state code largely adopts the wind sea definition of the Douglas Sea Scale. The direction from which the swell is coming should be recorded, in engineering applications, sea states are often characterized by the following two parameters, The significant wave height H1/3 — the mean wave height of the one third highest waves. The sea state is in addition to these two parameters also described by the wave spectrum S which is a function of a wave height spectrum S, some wave height spectra are listed below. The dimension of the spectrum is =, and many interesting properties about the sea state can be found from the spectrum. In addition to the term wave statistics presented above, long term sea state statistics are often given as a joint frequency table of the significant wave height. From the long and short term statistical distributions it is possible to find the extreme values expected in the life of a ship. Surviving the once in 100 years or once in 1000 years sea state is a demand for design of ships. Beaufort scale Cross sea Douglas Sea Scale Bowditch, Nathaniel, American Practical Navigator,9, United States Hydrographic Office, OCLC31033357 Faltinsen, O. M. Sea Loads on Ships and Offshore Structures, ISBN 0-521-45870-6

27.
Seiche
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A seiche is a standing wave in an enclosed or partially enclosed body of water. Seiches and seiche-related phenomena have been observed on lakes, reservoirs, swimming pools, bays, harbours, the key requirement for formation of a seiche is that the body of water be at least partially bounded, allowing the formation of the standing wave. The term was promoted by the Swiss hydrologist François-Alphonse Forel in 1890, the word originates in a Swiss French dialect word that means to sway back and forth, which had apparently long been used in the region to describe oscillations in alpine lakes. Seiches are often imperceptible to the eye, and observers in boats on the surface may not notice that a seiche is occurring due to the extremely long wavelengths. The effect is caused by resonances in a body of water that has been disturbed by one or more of a number of factors, most often meteorological effects, seismic activity or by tsunamis. Gravity always seeks to restore the surface of a body of liquid water. Vertical harmonic motion results, producing an impulse that travels the length of the basin at a velocity that depends on the depth of the water, the impulse is reflected back from the end of the basin, generating interference. Repeated reflections produce standing waves with one or more nodes, or points, the frequency of the oscillation is determined by the size of the basin, its depth and contours, and the water temperature. The longest natural period of a seiche is the associated with the fundamental resonance for the body of water—corresponding to the longest standing wave. Higher order harmonics are also observed, the period of the second harmonic will be half the natural period, the period of the third harmonic will be a third of the natural period, and so forth. Seiches have been observed on lakes and seas. The key requirement is that the body of water be partially constrained to allow formation of standing waves, regularity of geometry is not required, even harbours with exceedingly irregular shapes are routinely observed to oscillate with very stable frequencies. Low rhythmic seiches are almost always present on larger lakes and they are usually unnoticeable among the common wave patterns, except during periods of unusual calm. Harbours, bays, and estuaries are often prone to small seiches with amplitudes of a few centimetres, among other lakes well known for their regular seiches is New Zealands Lake Wakatipu, which varies its surface height at Queenstown by 20 centimetres in a 27-minute cycle. Seiches can also form in semi-enclosed seas, the North Sea often experiences a lengthwise seiche with a period of about 36 hours, the National Weather Service issues low water advisories for portions of the Great Lakes when seiches of 2 feet or greater are likely to occur. These can lead to extreme seiches of up to 5 m between the ends of the lake, the effect is similar to a storm surge like that caused by hurricanes along ocean coasts, but the seiche effect can cause oscillation back and forth across the lake for some time. The same storm system caused the 1995 seiche on Lake Superior produced a similar effect in Lake Huron. On Lake Michigan, eight fishermen were swept away from piers at Montrose and North Avenue Beaches, Lakes in seismically active areas, such as Lake Tahoe in California/Nevada, are significantly at risk from seiches

28.
Significant wave height
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In physical oceanography, the significant wave height is defined traditionally as the mean wave height of the highest third of the waves. Nowadays it is defined as four times the standard deviation of the surface elevation – or equivalently as four times the square root of the zeroth-order moment of the wave spectrum. The symbol Hm0 is usually used for that latter definition, the significant wave height may thus refer to Hm0 or H1/3, the difference in magnitude between the two definitions is only a few percent. The original definition resulted from work by the oceanographer Walter Munk during World War II, the significant wave height was intended to mathematically express the height estimated by a trained observer. It is commonly used as a measure of the height of ocean waves, significant wave height, scientifically represented as Hs or Hsig, is an important parameter for the statistical distribution of ocean waves. The most common waves are less in height than Hs and this implies that encountering the significant wave is not too frequent. However, statistically, it is possible to encounter a wave that is higher than the significant wave. Generally, the distribution of the individual wave heights is well approximated by a Rayleigh distribution. However, in changing conditions, the disparity between the significant wave height and the largest individual waves might be even larger. Other statistical measures of the wave height are also widely used, the RMS wave height, which is defined as square root of the average of the squares of all wave heights, is approximately equal to Hs divided by 1.4. For example, according to the Irish Marine Institute, … at midnight on 9/12/2007 a record significant wave height was recorded of 17. 2m at with a period of 14 seconds. The maximum ever measured wave height from a satellite is 20. 1m during a North Atlantic storm in 2011, the World Meteorological Organization stipulates that certain countries are responsible for providing weather forecasts for the worlds oceans. These respective countries meteorological offices are called Regional Specialized Meteorological Centers, in their weather products, they give ocean wave height forecasts in significant wave height. In the United States, NOAAs National Weather Service is the RSMC for a portion of the North Atlantic, the Ocean Prediction Center and the Tropical Prediction Centers Tropical Analysis and Forecast Branch issue these forecasts. RSMCs use wind-wave models as tools to predict the sea conditions. In the U. S. NOAAs WAVEWATCH III model is used heavily, a significant wave height is also defined similarly, from the wave spectrum, for the different systems that make up the sea. We then have a significant wave height for the wind-sea or for a particular swell

29.
Soliton
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In mathematics and physics, a soliton is a self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium, solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell who observed a solitary wave in the Union Canal in Scotland and he reproduced the phenomenon in a wave tank and named it the Wave of Translation. A single, consensus definition of a soliton is difficult to find, more formal definitions exist, but they require substantial mathematics. Moreover, some use the term soliton for phenomena that do not quite have these three properties. Dispersion and non-linearity can interact to produce permanent and localized wave forms, consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of different frequencies. Since glass shows dispersion, these different frequencies will travel at different speeds, however, there is also the non-linear Kerr effect, the refractive index of a material at a given frequency depends on the lights amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the effect, and the pulses shape will not change over time. See soliton for a detailed description. The soliton solutions are obtained by means of the inverse scattering transform. The mathematical theory of equations is a broad and very active field of mathematical research. Some types of tidal bore, a phenomenon of a few rivers including the River Severn, are undular. Other solitons occur as the internal waves, initiated by seabed topography. Atmospheric solitons also exist, such as the Morning Glory Cloud of the Gulf of Carpentaria, the recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons. A topological soliton, also called a defect, is any solution of a set of partial differential equations that is stable against decay to the trivial solution. Soliton stability is due to constraints, rather than integrability of the field equations. Thus, the differential equation solutions can be classified into homotopy classes, there is no continuous transformation that will map a solution in one homotopy class to another

30.
Stokes boundary layer
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In fluid dynamics, the Stokes boundary layer, or oscillatory boundary layer, refers to the boundary layer close to a solid wall in oscillatory flow of a viscous fluid. Or, it refers to the case of an oscillating plate in a viscous fluid at rest. In turbulent flow, this is named a Stokes boundary layer. The thickness of the boundary layer is called the Stokes boundary-layer thickness. This observation is also valid for the case of a turbulent boundary layer, outside the Stokes boundary layer – which is often the bulk of the fluid volume – the vorticity oscillations may be neglected. To good approximation, the velocity oscillations are irrotational outside the boundary layer. This significantly simplifies the solution of these problems, and is often applied in the irrotational flow regions of sound waves. The oscillating flow is assumed to be uni-directional and parallel to the plane wall, the only non-zero velocity component is called u and is in the x-direction parallel to the oscillation direction. Moreover, since the flow is taken to be incompressible, the velocity component u is only a function of time t, the Reynolds number is taken small enough for the flow to be laminar. In conclusion, the pressure forcing ∂p/∂x can only be a function of time t, the only non-zero component of the vorticity vector is the one in the direction perpendicular to x and z, called ω and equal to, ω = ∂ u ∂ z. Taking the z-derivative of the equation, ω has to satisfy ∂ ω ∂ t = ν ∂2 ω ∂ z 2. As usual for the vorticity dynamics, the pressure out of the vorticity equation. Harmonic motion of a rigid plate – moving parallel to its plane – will result in the fluid near the plate being dragged with the plate. Suppose the motion of the plate is u 0 = U0 cos ⁡, with U0 the velocity amplitude of the plate motion, and Ω the angular frequency of the motion. The plate, located at z =0, forces the fluid adjacent to have the same velocity u1 resulting in the no-slip condition. Far away from the plate, for z → ∞, the velocity u1 approaches zero, such an equation is called a one-dimensional heat equation or diffusion equation. As a result, the solution for the velocity is u 1 = U0 e − κ z cos ⁡ with κ = Ω2 ν. Here, κ is a kind of wavenumber in the z-direction, at a distance δ from the plate, the velocity amplitude has been reduced to e−2π ≈0.002 times its value U0 at the plate surface

31.
Stokes drift
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For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation. More generally, the Stokes drift velocity is the difference between the average Lagrangian flow velocity of a parcel, and the average Eulerian flow velocity of the fluid at a fixed position. This nonlinear phenomenon is named after George Gabriel Stokes, who derived expressions for this drift in his 1847 study of water waves. The Stokes drift is the difference in end positions, after an amount of time. The end position in the Lagrangian description is obtained by following a specific fluid parcel during the time interval, the Stokes drift velocity equals the Stokes drift divided by the considered time interval. Often, the Stokes drift velocity is referred to as Stokes drift. Stokes drift may occur in all instances of oscillatory flow which are inhomogeneous in space, for instance in water waves, tides and atmospheric waves. In the Lagrangian description, fluid parcels may drift far from their initial positions, as a result, the unambiguous definition of an average Lagrangian velocity and Stokes drift velocity, which can be attributed to a certain fixed position, is by no means a trivial task. However, such a description is provided by the Generalized Lagrangian Mean theory of Andrews. The Stokes drift is important for the transfer of all kind of materials. Further the Stokes drift is important for the generation of Langmuir circulations, for nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated. Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the time t = t0. But also other ways of labeling the fluid parcels are possible, different definitions of the average may be used, depending on the subject of study, see ergodic theory, time average, space average, ensemble average and phase average. Now, the Stokes drift velocity ūS equals u ¯ S = u ¯ L − u ¯ E, in many situations, the mapping of average quantities from some Eulerian position x to a corresponding Lagrangian position α forms a problem. Since a fluid parcel with label α traverses along a path of many different Eulerian positions x, a mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the Generalized Lagrangian Mean by Andrews and McIntyre. Here the last term describes the Stokes drift 12 k u ^2 / ω, the Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. Further, the Stokes drift velocity decays exponentially with depth, at a depth of a wavelength, z = -¼ λ, it is about 4% of its value at the mean free surface

32.
Stokes wave
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In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. Stokes wave theory is of practical use for waves on intermediate. It is used in the design of coastal and offshore structures, the wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude, in such shallow water, a cnoidal wave theory often provides better periodic-wave approximations. While, in the sense, Stokes wave refers to progressive periodic waves of permanent form. The examples below describe Stokes waves under the action of gravity in case of wave motion. The phase speed increases with increasing non-linearity ka of the waves, the wave height H, being the difference between the surface elevation η at a crest and a trough, is, H =2 a. Note that the second- and third-order terms in the velocity potential Φ are zero, only at fourth order contributions deviating from first-order theory – i. e. Airy wave theory – appear. Up to third order the orbital velocity field u = ∇Φ consists of a motion of the velocity vector at each position. As a result, the elevation of deep-water waves is to a good approximation trochoidal. Stokes further observed, that although the orbital velocity field consists of a circular motion at each point. This is due to the reduction of the velocity amplitude at increasing depth below the surface and this Lagrangian drift of the fluid parcels is known as the Stokes drift. Observe that for finite depth the velocity potential Φ contains a drift in time. Both this temporal drift and the term in Φ vanish for deep-water waves. The ratio S of the free-surface amplitudes at second or and first order – according to Stokes second-order theory – is, in deep water, for large kh the ratio S has the asymptote lim k h → ∞ S =12 k a. Here U is the Ursell parameter, for long waves of small height H, i. e. U ≪ 32π2/3 ≈100, second-order Stokes theory is applicable. Otherwise, for long waves of appreciable height H a cnoidal wave description is more appropriate. According to Hedges, fifth-order Stokes theory is applicable for U <40, for Stokes waves under the action of gravity, the third-order dispersion relation is – according to Stokes first definition of celerity, ω2 = + O, with σ = tanh k h

33.
Swell (ocean)
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A swell, in the context of an ocean, sea or lake, is a series of mechanical waves that propagate along the interface between water and air and so they are often referred to as surface gravity waves. These series of gravity waves are not generated by the immediate local wind, instead by distant weather systems. This is the definition of a swell as opposed to a locally generated wind wave. More generally, a swell consists of wind-generated waves that are not—or are hardly—affected by the wind at that time. Swell wavelength, also, varies from event to event, occasionally, swells which are longer than 700 m occur as a result of the most severe storms. Swell direction is the direction from which the swell is coming and it is measured in degrees, and often referred to in general directions, such as a NNW or SW swell. Large breakers one observes on a beach may result from distant weather systems over a fetch of ocean, further exposure to that specific wind could only cause a loss of energy due to the breaking of wave tops and formation of whitecaps. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer would estimate from visual observation of a sea state, given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. Sea water wave is generated by many kinds of such as Seismic events, gravity. The generation of wave is initiated by the disturbances of cross wind field on the surface of the sea water. Two major mechanisms of surface wave formation by winds and other sources of wave formation can explain the generation of wind waves. 1) Starts from Fluctuations of wind, the wind wave formation on water surface by wind is started by a distribution of normal pressure acting on the water from the wind. By the mechanism developed by O. M and this pressure fluctuation arise normal and tangential stresses to the surface water, and generates wave behavior on the water surface. Since the wind profile Ua is logarithmic to the water surface and this relations show the wind flow transferring its kinetic energy to the water surface at their interface, and arises wave speed, c. For example, If we suppose a very flat sea surface, turbulent wind flows form random pressure fluctuations at the sea surface

34.
Trochoidal wave
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In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this solution is an inverted trochoid – with sharper crests. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863, the flow field associated with the trochoidal wave is not irrotational, it has vorticity. The vorticity is of such a strength and vertical distribution that the trajectories of the fluid parcels are closed circles. This is in contrast with the experimental observation of Stokes drift associated with the wave motion. Also the phase speed is independent of the waves amplitude, unlike other nonlinear wave-theories. For these reasons – as well as for the fact that solutions for finite fluid depth are lacking – trochoidal waves are of limited use for engineering applications, in computer graphics, the rendering of realistic-looking ocean waves can be done by use of so-called Gerstner waves. This is a multi-component and multi-directional extension of the traditional Gerstner wave, the Lagrangian coordinates label the fluid parcels, with = the centres of the circular orbits – around which the corresponding fluid parcel moves with constant speed c exp ⁡. Further k =2 π / λ is the wavenumber, while c is the speed with which the wave propagates in the x -direction. The phase speed satisfies the relation, c 2 = g k, which is independent of the wave nonlinearity. The free surface is a line of constant pressure, and is found to correspond with a line b = b s, for b s =0 the highest waves occur, with a cusp-shaped crest. Note that the highest Stokes wave has a crest angle of 120°, the wave height of the trochoidal wave is H = exp ⁡. The wave is periodic in the x -direction, with wavelength λ, and also periodic in time with period T = λ / c =2 π λ / g. The vorticity ϖ under the wave is, ϖ = −2 k c e 2 k b 1 − e 2 k b, varying with Lagrangian elevation b. For the classical Gerstner wave the fluid motion exactly satisfies the nonlinear, incompressible, however, the extended Gerstner waves do in general not satisfy these flow equations exactly. This description of the ocean can be programmed very efficiently by use of the fast Fourier transform, moreover, the resulting ocean waves from this process look realistic, as a result of the nonlinear deformation of the free surface, sharper crests and flatter troughs. The mathematical description of the free-surface in these Gerstner waves can be as follows, the coordinates are denoted as x and z

35.
Tsunami
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A tsunami or tidal wave, also known as a seismic sea wave, is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Unlike normal ocean waves which are generated by wind, or tides which are generated by the pull of the Moon and Sun. Tsunami waves do not resemble normal undersea currents or sea waves, Tsunamis generally consist of a series of waves with periods ranging from minutes to hours, arriving in a so-called internal wave train. Wave heights of tens of metres can be generated by large events, numerous terms are used in the English language to describe waves created in a body of water by the displacement of water, however, none of the terms in frequent use are entirely accurate. The term tsunami, meaning harbour wave in literal translation, comes from the Japanese 津波, while not entirely accurate, as tsunami are not restricted to harbours, tsunami is currently the term most widely accepted by geologists and oceanographers. Tsunami are sometimes referred to as tidal waves and this once-popular term derives from the most common appearance of tsunami, which is that of an extraordinarily high tidal bore. Although the meanings of tidal include resembling or having the form or character of the tides, use of the tidal wave is discouraged by geologists. The term seismic sea wave also is used to refer to the phenomenon, prior to the rise of the use of the term tsunami in English-speaking countries, scientists generally encouraged the use of the term seismic sea wave rather than tidal wave. The Sumatran region is not unused to tsunamis either, with earthquakes of varying magnitudes regularly occurring off the coast of the island, Tsunamis are an often underestimated hazard in the Mediterranean Sea and parts of Europe. The tsunami claimed more than 123,000 lives in Sicily, the Storegga Slide in the Norwegian sea and some examples of tsunamis affecting the British Isles refer to landslide and meteotsunamis predominantly and less to earthquake-induced waves. The cause, in my opinion, of this phenomenon must be sought in the earthquake, at the point where its shock has been the most violent the sea is driven back, and suddenly recoiling with redoubled force, causes the inundation. Without an earthquake I do not see how such an accident could happen, the principal generation mechanism of a tsunami is the displacement of a substantial volume of water or perturbation of the sea. This displacement of water is attributed to either earthquakes, landslides, volcanic eruptions, glacier calvings or more rarely by meteorites. The waves formed in this way are then sustained by gravity, tides do not play any part in the generation of tsunamis. Tsunami can be generated when the sea floor abruptly deforms and vertically displaces the overlying water and they grow in height when they reach shallower water, in a wave shoaling process described below. A tsunami can occur in any state and even at low tide can still inundate coastal areas. On April 1,1946, the 8.6 Mw Aleutian Islands earthquake occurred with a maximum Mercalli intensity of VI and it generated a tsunami which inundated Hilo on the island of Hawaii with a 14-metre high surge. Between 165 and 173 were killed, the area where the earthquake occurred is where the Pacific Ocean floor is subducting under Alaska

36.
Megatsunami
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A megatsunami is a term used for a very large wave created by a large, sudden displacement of material into a body of water. Megatsunamis have quite different features from other, more types of tsunamis. Most tsunamis are caused by tectonic activity and therefore occur along plate boundaries and as a result of earthquake and rise or fall in the sea floor. Ordinary tsunamis have shallow waves out at sea, and the water piles up to a height of up to about 10 metres as the sea floor becomes shallow near land. By contrast, megatsunamis occur when a large amount of material suddenly falls into water or anywhere near water. Modern megatsunamis include the one associated with the 1883 eruption of Krakatoa, the 1958 Lituya Bay megatsunami, prehistoric examples include the Storegga Slide, and the Chicxulub, Chesapeake Bay and Eltanin meteor impacts. A megatsunami is a large wave due to displacement of a body of water—with an initial wave amplitude measured in several tens, hundreds. Normal tsunamis generated at sea result from movement of the sea floor and they have a small wave height offshore, are very long, and generally pass unnoticed at sea, forming only a slight swell usually of the order of 30 cm above the normal sea surface. When they reach land, the height increases dramatically as the base of the wave pushes the water column above it upwards. By contrast, megatsunamis are caused by giant landslides and other impact events and this could also refer to a meteorite hitting an ocean. If the landslide or impact occurs in a body of water, as happened at the Vajont Dam and Lituya Bay then the water may be unable to disperse. In this analogy, a megatsunami would be similar to dropping a large rock from a considerable height into the tub, at one end, causing water to splash up and out. Geologists searching for oil in Alaska in 1953 observed that in Lituya Bay, rather, there was a band of younger trees closer to the shore. Forestry workers, glaciologists, and geographers call the boundary between these bands a trim line, trees just above the trim line showed severe scarring on their seaward side, whilst those from below the trim line did not. The scientists hypothesized that there had been a large wave or waves in the deep inlet. Because this is a recently deglaciated fjord with steep slopes and crossed by a major fault, one possibility was a landslide-generated tsunami. On 9 July 1958, a 7.8 Mw strike-slip earthquake in southeast Alaska caused 90 million tonnes of rock, over the trees in their fishing boat. They were washed back into the bay and both survived, neither water drainage from a lake, nor landslide, nor the force of the earthquake itself led to the megatsunami, although all of these may have contributed

37.
Undertow (water waves)
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In physical oceanography, undertow is the average under-current which is moving offshore when waves are approaching a shore. Undertow is a necessary and universal feature, it is a return flow compensating for the average transport of water by the waves in the zone above the wave troughs. The undertows flow velocities are generally strongest in the zone, where the water is shallow. In popular usage, the word undertow is often misapplied to rip currents, an undertow occurs everywhere underneath shore-approaching waves, whereas rip currents are localized narrow offshore currents occurring at certain locations along the coast. Unlike undertow, rip currents are strong at the surface, an undertow is a steady, offshore-directed compensation flow, which occurs below waves near the shore. Physically, nearshore, the mass flux between wave crest and trough is onshore directed. This mass transport is localized in the part of the water column. To compensate for the amount of water being transported towards the shore and this flow – the undertow – affects the nearshore waves everywhere, unlike rip currents localized at certain positions along the shore. The term undertow is used in scientific coastal oceanography papers, the distribution of flow velocities in the undertow over the water column is important as it strongly influences the on- or offshore transport of sediment. Outside the surf zone there is a near-bed onshore-directed sediment transport induced by Stokes drift, in the surf zone, strong undertow generates a near-bed offshore sediment transport. These antagonistic flows may lead to sand bar formation where the flows converge near the breaking point. An exact relation for the flux of a nonlinear periodic wave on an inviscid fluid layer was established by Levi-Civita in 1924. The positive flow direction of u ¯ is in the propagation direction. Since in general the potential energy E p is much easier to measure than the kinetic energy, so u ¯ ≈ −18 g H2 c h. For irregular waves the wave height is the root-mean-square wave height H rms ≈8 σ. The potential energy is E p =12 ρ g σ2 and E w ≈ ρ g σ2, the distribution of the undertow velocity over the water depth is a topic of ongoing research. In popular usage, the word undertow is used correctly. This misconception stems from a lack of knowledge about water currents

38.
Ursell number
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In fluid dynamics, the Ursell number indicates the nonlinearity of long surface gravity waves on a fluid layer. This dimensionless parameter is named after Fritz Ursell, who discussed its significance in 1953, so the Ursell parameter U is the relative wave height H / h times the relative wavelength λ / h squared. For long waves with small Ursell number, U ≪32 π2 /3 ≈100, otherwise a non-linear theory for fairly long waves – like the Korteweg–de Vries equation or Boussinesq equations – has to be used. The parameter, with different normalisation, was introduced by George Gabriel Stokes in his historical paper on surface gravity waves of 1847. Dingemans, M. W. Water wave propagation over uneven bottoms

39.
Wave base
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The wave base, in physical oceanography, is the maximum depth at which a water waves passage causes significant water motion. For water depths deeper than the base, bottom sediments. In seawater, the particles are moved in a circular orbital motion when a wave passes. The radius of the circle of motion for any water molecule decreases exponentially with increasing depth. The wave base, which is the depth of influence of a wave, is about half the wavelength. At depths greater than half the wavelength, the motion is less than 4% of its value at the water surface. For example, in a pool of water 1 metre deep, in the same pool, a wave with a wavelength of 0.5 metres would not be able to cause water movement on the bottom. There are typically two wave bases, the fair weather wave base and the wave base. The fair weather wave base refers to the depth beneath the waves under normal conditions, the storm wave base refers to the depths beneath storm-driven waves and can be much deeper. The portion of the seafloor that is agitated by storm-driven wave action is known as the Lower shoreface. Upper shoreface — above wave base Lower shoreface — below wave base Airy wave theory Dispersion Waves and shallow water

Coupling data collected by NASA/JPL by several different satellite-borne sensors, researchers have been able to "break through" the ocean's surface to detect "Meddies" – super-salty warm-water eddies that originate in the Mediterranean Sea and then sink more than a half-mile underwater in the Atlantic Ocean. The Meddies are shown in red in this scientific figure.

In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating …

Image from Cursus seu Mundus Mathematicus (1674) of C.F.M. Dechales, showing how a cannonball should deflect to the right of its target on a rotating Earth, because the rightward motion of the ball is faster than that of the tower.

Image from Cursus seu Mundus Mathematicus (1674) of C.F.M. Dechales, showing how a ball should fall from a tower on a rotating Earth. The ball is released from F. The top of the tower moves faster than its base, so while the ball falls, the base of the tower moves to I, but the ball, which has the eastward speed of the tower's top, outruns the tower's base and lands further to the east at L.

A carousel is rotating counter-clockwise. Left panel: a ball is tossed by a thrower at 12:00 o'clock and travels in a straight line to the center of the carousel. While it travels, the thrower circles in a counter-clockwise direction. Right panel: The ball's motion as seen by the thrower, who now remains at 12:00 o'clock, because there is no rotation from their viewpoint.

This low-pressure system over Iceland spins counterclockwise due to balance between the Coriolis force and the pressure gradient force.

The phases of an ocean surface wave: 1. Wave Crest, where the water masses of the surface layer are moving horizontally in the same direction as the propagating wave front. 2. Falling wave. 3. Trough, where the water masses of the surface layer are moving horizontally in the opposite direction of the wave front direction. 4. Rising wave.

Wave motion on the interface between two layers of inviscid homogeneous fluids of different density, confined between horizontal rigid boundaries (at the top and bottom). The motion is forced by gravity. The upper layer has mean depth h‘ and density ρ‘, while the lower layer has mean depth h and density ρ. The wave amplitude is a, the wavelength is denoted by λ.

In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and …

A simulation with a Boussinesq-type wave model of nearshore waves travelling towards a harbour entrance. The simulation is with the BOUSS-2D module of SMS.

Periodic waves in the Boussinesq approximation, shown in a vertical cross section in the wave propagation direction. Notice the flat troughs and sharp crests, due to the wave nonlinearity. This case (drawn on scale) shows a wave with the wavelength equal to 39.1 m, the wave height is 1.8 m (i.e. the difference between crest and trough elevation), and the mean water depth is 5 m, while the gravitational acceleration is 9.81 m/s2.

The Ekman layer is the layer in a fluid where there is a force balance between pressure gradient force, Coriolis force …

The Ekman layer is the layer in a fluid where the flow is the result of a balance between pressure gradient, Coriolis and turbulent drag forces. In the picture above, the wind blowing North creates a surface stress and a resulting Ekman spiral is found below it in the column of water.

Three views of the wind-driven Ekman layer at the surface of the ocean in the Northern Hemisphere. The geostrophic velocity is zero in this example.